Compensating Masks, Multi-Optical Systems Using the Masks, and Methods of Compensating for 3-D Mask Effect Using the Same

ABSTRACT

Provided are a compensating mask, a multi-optical system using the compensating mask, and a method of compensating for a 3-dimensional (3-D) mask effect using the compensating mask. Methods of compensating for a 3-D mask effect using a compensating mask may include generating a first kernel corresponding to a normal mask used for forming a minute pattern, generating a second kernel corresponding to a compensating mask, mixing the first kernel corresponding to the normal mask with the second kernel corresponding to the compensating mask, and generating a multi-optical system kernel corresponding to mixing the first kernel and the second kernel.

CROSS-REFERENCE TO RELATED PATENT APPLICATION

This U.S. non-provisional patent application claims priority under 35U.S.C. §120 as a divisional of U.S. patent application Ser. No.11/925,014, filed Oct. 26, 2007, which in turn claims priority under 35U.S.C. §119 to Korean Patent Application No. 10-2006-0107947, filed onNov. 2, 2006, in the Korean Intellectual Property Office, the disclosureof which is incorporated herein in its entirety by reference.

BACKGROUND OF THE INVENTION

The present invention relates to apparatus, systems and methods forfabricating a semiconductor device, and more particularly to, masking ina semiconductor fabricating process.

Semiconductors have become highly integrated and, as such, the componentdimensions have become increasingly minute. Thus, there may be increaseddemand for mask pattern resolution to accommodate pattern dimensionsthat may be less than the wavelength of light used in an exposureapparatus.

Optical proximity correction (OPC) technology may be used to correct theshape of a mask pattern that may experience deformation caused by anoptical proximity effect for a pattern having a shorter line width thanthe wavelength of light. Examples of OPC technology include, forexample, model-based OPC and rule-based OPC. Model-based OPC can be moreeasily applied to various layouts than in rule-based OPC.

A current sub-50 nm device process may need a pattern scale of a mask ofabout 4 times, i.e. below 200 nm. However, a mask of this size may havea narrow structure, which may not be easily penetrated by an ArF laserused as a light source. As a result, internal scattering, mask inducedpolarization, and reflection loss due to pellicle may cause a3-dimensional (3-D) mask effect.

Rigorous simulations may be performed to compensate for the 3-D maskeffect, such as finite difference time domain (FDTD) analysis, rigorouscoupled wave analysis (RCWA), and time-domain electromagnetic massivelyparallel evaluation of scattering from topography (TEMPEST) as a kind ofFDTD. However, the rigorous simulations may be difficult to apply whenOPC is performed over a large area. Further, such rigorous simulationsmay not give significantly better results than conventional simulations.

Reference is made to FIG. 1, which schematically illustrates a typicalscanner system that may be used in photolithography. The scanner systemincludes a light source 10, an illumination lens 20, a mask 30, and aprojection lens 40, among other components. An illumination pupil 22 maybe formed on the illumination lens 20, and an imaging pupil 42 may beformed on the projection lens 40 to correct a pupil surface. A wafer 50,on which a pattern is to be formed, may include a resist layer 54 coatedonto a silicon substrate 52. The wafer 50 may be placed under thescanner system and light may be irradiated onto the resist layer 54.Here, the mask 30 may include a light blocking layer 32 formed ofchromium (Cr) and a light transmitting layer 34 formed of quartz. Thesilicon substrate 52 may be a pure silicon substrate or a siliconsubstrate on which multiple material layers have been formed.

A conventional mask may approximate a thin mask, and OPC may beperformed without considering the mask's thickness. However, theaccuracy of a thin mask approximation may be lower when the feature sizeor pattern size of the mask approaches the wavelength of the lightsource, e.g. an ArF wavelength. In other words, in a current thin maskapproximation method, mask features are effective at above 2.5× thelight source wavelength. Three-dimensional (3-D) mask effects, however,may occur below that size. In this regard, it may be difficult to adoptthe thin mask approximation method.

Reference is made to FIG. 2A, which is a cross-sectional view of aportion of a mask 30. The mask 30 may include a light transmitting layer34 formed of glass and a light blocking layer 32 formed of chromium (Cr)underneath the light transmitting layer 34. Patterns may be formed inthe light blocking layer 32 to form openings of width W. Although oneopening is illustrated by way of example, a mask 30 may include multipleopenings corresponding to a pattern. Such an opening may generally havea width of about four times the pitch or line width of a wafer pattern.

Reference is now made to FIG. 2B, which is a graph of percent differenceof a critical dimension (CD) of a wafer as a function of the width of anopening of a light blocking layer 32, as illustrated in FIG. 2A. Thex-axis denotes the line width of a wafer pattern, and the y-axis denotesthe percent difference of the CD. By way of example, the light sourcehas a wavelength of 193 nm, the numerical aperture (NA) is 0.75, and thecoherence coefficient σ is 0.35. Reference character M denotes the sizefactor of the opening of the mask 30 with respect to the line width ofthe wafer. Here, the factor M is “4.” As illustrated, the difference ofthe CD, which may also be referred to as error, increases abruptly atline widths below 150 nm. This can be attributed to inaccuracies in OPCwhich does not account for a 3-D mask effect as described above.

SUMMARY OF THE INVENTION

The present invention provides a mask that includes a compensating maskin a thin mask pattern configured to compensate for a 3-D(3-dimensional) effect caused by the thickness of a normal mask used forforming a minute pattern. Some embodiments of a compensating mask mayinclude a light transmitting layer formed of a light transmittingmaterial; and a light blocking layer patterned underneath the lighttransmitting layer and formed of a light blocking material.

In some embodiments, a weighting function of the compensating maskcorresponds to a scattering coefficient α of a near field image of thenormal mask. In some embodiments, the weighting function is used to mixa first kernel of the normal mask with a second kernel of thecompensating mask so as to generate a multi-optical system kernel. Insome embodiments, the multi-optical system kernel is applied on a waferand a result measured on the wafer is compared with a desired patternspecification to calibrate the scattering coefficient α and theweighting function.

Some embodiments provide that standard optical parameters and a pupilsurface function of a projection lens positioned under a mask of anoptical system are calibrated to generate the first kernel of the normalmask and the second kernel of the compensating mask. In someembodiments, the standard optical parameters include an illuminationcondition, an NA (numeral aperture), and a wavelength, and the pupilsurface function includes a magnitude and a phase of light as factors.

Some embodiments of the present invention include computer programproducts that may include a computer usable storage medium havingcomputer-readable program code embodied in the medium, the computerreadable program code configured to generate a kernel of thecompensating mask of embodiments described herein.

Some embodiments of the present invention include a multi-optical systemthat may include a light source configured to irradiate light, anillumination lens configured to calibrate the irradiated light, and anormal mask configured to transfer a predetermined minute pattern onto awafer via the calibrated light. Such embodiments may further include acompensating mask formed in a thin mask pattern to compensate for a 3-D(3-dimensional) effect caused by the thickness of a normal mask, thecompensating mask including a light transmitting layer formed of a lighttransmitting material and a light blocking layer patterned underneaththe light transmitting layer and formed of a light blocking material anda projection lens focusing the light transmitted through the mask ontothe wafer.

In some embodiments, an image pupil surface is formed on the projectionlens. In some embodiments, a first kernel corresponding to the normalmask is mixed with a second kernel corresponding to the compensatingmask in the multi-optical system to generate a multi-optical systemkernel. In some embodiments, standard optical parameters and a pupilsurface function of the projection lens positioned under a mask of anoptical system are calibrated to generate the kernels of the normal andthe compensating mask. In some embodiments, the multi-optical systemkernel is formed according to a weighting function of the second kernelcorresponding to the compensating mask.

In some embodiments, the weighting function corresponds to a scatteringcoefficient α of a near field image of the normal mask. In someembodiments, the multi-optical system kernel is applied on a wafer, anda result measured on the wafer is compared with a desired patternspecification to calibrate the scattering coefficient α and theweighting function. In some embodiments, the pupil surface function iscalibrated according to the weighting function to generate themulti-optical system kernel.

Some embodiments of the present invention include methods ofcompensating for a 3-D mask effect using a compensating mask. Someembodiments of such methods may include generating a first kernelcorresponding to a normal mask used for forming a minute pattern,generating a second kernel corresponding to a compensating mask andmixing the first kernel corresponding to the normal mask with the secondkernel corresponding to the compensating mask. Such methods may furtherinclude generating a multi-optical system kernel corresponding to mixingthe first kernel and the second kernel.

Some embodiments may include calibrating standard optical parameters anda pupil surface function of a projection lens positioned under a mask ofan optical system to generate the first kernel and the second kernelcorresponding to the normal and the compensating masks, respectively. Insome embodiments, the standard optical parameters include anillumination condition, an NA (numerical aperture), and a wavelength,and wherein the pupil surface function comprises a magnitude and a phaseof light as factors.

Some embodiments may include forming the multi-optical system kernelaccording to a weighting function of the kernel of the compensatingmask. Some embodiments include applying the multi-optical system kernelon a wafer and comparing a result measured on the wafer a desiredpattern specification to calibrate the weighting function. In someembodiments, the weighting function corresponds to a scatteringcoefficient α of a near field image of the normal mask.

In some embodiments, generating the multi-optical system kernel includescalibrating the pupil surface function according to the weightingfunction. In some embodiments, generating the multi-optical systemkernel includes inputting a predetermined value to the scatteringcoefficient α to obtain the weighting function, applying the weightingfunction to mix the first kernel corresponding to the normal with thesecond kernel corresponding to the compensating mask, and applying themulti-optical system kernel on a wafer to measure a result. Suchembodiments may further include comparing the measured result with adesired pattern specification, calibrating the scattering coefficient αaccording to the comparison to obtain a new weighting function, andapplying the new weighting function to mix the first kernelcorresponding to the normal mask with the second kernel corresponding tothe compensating mask.

Some embodiments include performing OPC (optical proximity correction)before generating the kernel of the normal mask and the kernel of thecompensating mask. Some embodiments include reflecting restrictionconditions according to a specific pattern of the wafer after generatingthe multi-optical system kernel. Some embodiments include performingcalibration so that the multi-optical system is applied to a defocused3-D model after generating the multi-optical system kernel. In someembodiments, the methods may be applied to a mask scale of 200 nm orless.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a side cross-sectional diagram that schematically illustratesa scanner system for photolithography.

FIGS. 2A and 2B are a cross-sectional view and a graph illustrating a3-dimensional (3-D) mask effect with respect to a reduction in a patternpitch of a mask.

FIGS. 3A through 3C are a partial cross-sectional view of a normal maskand graphs illustrating a light transmitting shape according to aposition of the normal mask according to some embodiments of the presentinvention.

FIG. 4 is a graph of a near field image according to a scatteringcoefficient for obtaining a weighting function of a compensating maskaccording to some embodiments of the present invention.

FIGS. 5A through 5C are graphs of pupil surface functions of a normalmask and a compensating mask and pupil surface functions corrected bythe normal and compensating masks, according to some embodiments of thepresent invention.

FIG. 6 is a composite graph showing the graphs of FIGS. 5A through 5Coverlaid according to some embodiments of the present invention.

FIG. 7 shows an effect of a compensating mask for compensating for a 3-Dmask effect in a multi-optical system according to some embodiments ofthe present invention.

FIG. 8 is a graph showing the profile of a transmission electric field(E-field) on pupil surfaces of normal and compensating masks and theprofile of a calibrated E-field according to some embodiments of a maskpattern in the multi-optical system of FIG. 7.

FIGS. 9A and 9B are photographs of pattern images of the multi-opticalsystem of FIG. 7 and pattern images on a wafer.

FIG. 10 is a flowchart illustrating operations for compensating for a3-D mask effect using a compensating mask according to some embodimentsof the present invention.

DETAILED DESCRIPTION OF THE INVENTION

The present invention now will be described more fully hereinafter withreference to the accompanying drawings, in which embodiments of theinvention are shown. However, this invention should not be construed aslimited to the embodiments set forth herein. Rather, these embodimentsare provided so that this disclosure will be thorough and complete, andwill fully convey the scope of the invention to those skilled in theart.

It will be understood that, although the terms first, second, etc. maybe used herein to describe various elements, these elements should notbe limited by these terms. These terms are only used to distinguish oneelement from another element. Thus, a first element discussed belowcould be termed a second element without departing from the scope of thepresent invention. In addition, as used herein, the singular forms “a”,“an” and “the” are intended to include the plural forms as well, unlessthe context clearly indicates otherwise. It also will be understoodthat, as used herein, the term “comprising” or “comprises” isopen-ended, and includes one or more stated elements, steps and/orfunctions without precluding one or more unstated elements, steps and/orfunctions. The term “and/or” includes any and all combinations of one ormore of the associated listed items.

It will also be understood that when an element is referred to as being“connected” to another element, it can be directly connected to theother element or intervening elements may be present. In contrast, whenan element is referred to as being “directly connected” to anotherelement, there are no intervening elements present. It will also beunderstood that the sizes and relative orientations of the illustratedelements are not shown to scale, and in some instances they have beenexaggerated for purposes of explanation. Like numbers refer to likeelements throughout.

In the figures, the dimensions of structural components, includinglayers and regions among others, are not to scale and may be exaggeratedto provide clarity of the concepts herein. It will also be understoodthat when a layer (or layer) is referred to as being ‘on’ another layeror substrate, it can be directly on the other layer or substrate, or canbe separated by intervening layers. Further, it will be understood thatwhen a layer is referred to as being ‘under’ another layer, it can bedirectly under, and one or more intervening layers may also be present.In addition, it will also be understood that when a layer is referred toas being ‘between’ two layers, it can be the only layer between the twolayers, or one or more intervening layers may also be present.

Unless otherwise defined, all terms (including technical and scientificterms) used herein have the same meaning as commonly understood by oneof ordinary skill in the art to which this invention belongs. It will befurther understood that terms, such as those defined in commonly useddictionaries, should be interpreted as having a meaning that isconsistent with their meaning in the context of the relevant art andwill not be interpreted in an idealized or overly formal sense unlessexpressly so defined herein.

The present invention may be embodied as apparatus, systems, methods,and/or computer program products. Accordingly, the present invention maybe embodied in hardware and/or in software (including firmware, residentsoftware, micro-code, etc.). Furthermore, the present invention may takethe form of a computer program product on a computer-usable orcomputer-readable storage medium having computer-usable orcomputer-readable program code embodied in the medium for use by or inconnection with an instruction execution system. In the context of thisdocument, a computer-usable or computer-readable medium may be anymedium that can contain, store, communicate, propagate, or transport theprogram for use by or in connection with the instruction executionsystem, apparatus, or device.

The computer-usable or computer-readable medium may be, for example butnot limited to, an electronic, magnetic, optical, electromagnetic,infrared, or semiconductor system, apparatus, device, or propagationmedium. More specific examples (a nonexhaustive list) of thecomputer-readable medium would include the following: an electricalconnection having one or more wires, a portable computer diskette, arandom access memory (RAM), a read-only memory (ROM), an erasableprogrammable read-only memory (EPROM or Flash memory), and a portablecompact disc read-only memory (CD-ROM). Note that the computer-usable orcomputer-readable medium could even be paper or another suitable mediumupon which the program is printed, as the program can be electronicallycaptured, via, for instance, optical scanning of the paper or othermedium, then compiled, interpreted, or otherwise processed in a suitablemanner, if necessary, and then stored in a computer memory.

The present invention is described herein with reference to flowchartand/or block diagram illustrations of methods, systems, and computerprogram products in accordance with exemplary embodiments of theinvention. It will be understood that each block of the flowchart and/orblock diagram illustrations, and combinations of blocks in the flowchartand/or block diagram illustrations, may be implemented by computerprogram instructions and/or hardware operations. These computer programinstructions may be provided to a processor of a general purposecomputer, a special purpose computer, or other programmable dataprocessing apparatus to produce a machine, such that the instructions,which execute via the processor of the computer or other programmabledata processing apparatus, create means for implementing the functionsspecified in the flowchart and/or block diagram block or blocks.

These computer program instructions may also be stored in a computerusable or computer-readable memory that may direct a computer or otherprogrammable data processing apparatus to function in a particularmanner, such that the instructions stored in the computer usable orcomputer-readable memory produce an article of manufacture includinginstructions that implement the function specified in the flowchartand/or block diagram block or blocks.

The computer program instructions may also be loaded onto a computer orother programmable data processing apparatus to cause a series ofoperational steps to be performed on the computer or other programmableapparatus to produce a computer implemented process such that theinstructions that execute on the computer or other programmableapparatus provide steps for implementing the functions specified in theflowchart and/or block diagram block or blocks.

Reference is now made to FIGS. 3A through 3C, which are a partialcross-sectional view of a normal mask and graphs illustrating a lighttransmitting shape according to a position of the normal mask accordingto some embodiments of the present invention. Referring to FIG. 3A,which is a partial cross-sectional view of a normal mask 100, the normalmask 100 includes a light transmitting layer 140 above a light blockinglayer 120. The light transmitting layer 140 may be formed of a varietyof light transmitting materials including, for example, glass and/orquartz, among others. The light blocking layer 120 includes patternshaving openings and may be formed of a metal including, for example,chromium (Cr). The normal mask 100 may be a phase shift mask (PSM),which is formed by alternately etching portions of the lighttransmitting layer 140 on adjacent openings of the light blocking layer120.

In some embodiments, an image formed in a portion adjacent the normalmask 100 after light has passed through the normal mask 100 may becalled a near field image 100. If the feature size and/or pattern sizeof the normal mask 100 is large, the normal mask 100 may accuratelyapproximate a thin mask. That is, the near field images of the normaland the thin mask may be similar to each other. However, as the maskfeature size decreases, a 3-D mask effect may occur due to diffractioncaused by the thickness of the normal mask 100. As a result, the normalmask 100 may not accurately approximate a thin mask. For example,high-order diffracted light beams may be reflected on a near fieldimage, which may vary depending on a position of the normal mask 100. Asa result, the normal mask 100 may not approximate the thin mask.

Z_(exc) denotes a portion on which light is incident, Z_(obj) denotes aportion of the opening for measuring a near field image of an etchedportion of the light transmitting layer 140, and Z_(obs) denotes aportion for measuring a near field image of a mask. Referring to FIG.3B, which schematically illustrates a left portion of an openingtransmitting light, light is incident on Z_(exc) and then diffused froman opening portion, i.e. the portion Z_(obs). Reference is now made toFIG. 3C, which illustrates an etched portion of the light transmittinglayer 140, i.e. the portion Z_(obj) of a right portion of the openingdiffusing light. If the pattern size of the mask is large, field imagesin the portions Z_(obj) and Z_(obs) are similar to each other. However,if the pattern size of the mask is small, light beams diffused from theportion Z_(obj) are diffracted when passing the opening. Thus, adifferent field image from a conventional field image is formed in theportion Z_(obs).

Reference is now made to FIG. 4, which is a graph of a near field imageaccording to a scattering coefficient for obtaining a weighting functionof a compensating mask according to some embodiments of the presentinvention. The near field image is expressed according to the scatteringcoefficient α. When the scattering coefficient α is infinite ∞, anoptimal near field image is formed. When the scattering coefficient α isfinite, an image is distorted by diffraction. In this regard, acompensating mask may be used to correct the distortion of the imagecaused by diffraction.

The compensating mask may be fabricated in the form of a thin mask. Akernel of the compensating mask may be generated and then mixed with akernel of a normal mask to generate a kernel of an optical mask. In someembodiments, a kernel is a model expressed as a transfer functioncorresponding to an optical proximity effect through OPC. A differencebetween a mask pattern and a pattern transferred onto a wafer throughthe mask pattern may be obtained in advance using the kernel, and thenthe mask pattern may be corrected according to the result of asimulation.

When a kernel of the compensating mask is mixed with the kernel of thenormal mask, a weight of the compensating mask should be included. Inthis manner, the extent to which a kernel of the compensating mask is tobe reflected to effectively exclude a 3-D mask effect of the normal maskmay be considered. Equation 1 of the weighting function expresses theweight of the compensating mask.

W _(j)=1/|k _(j) −k _(sim, pq)|^(α)  (1)

wherein α denotes the scattering coefficient, and k_(j) and k_(sim, pq)denote space frequency vectors respectively corresponding to theportions Z_(obj) and Z_(obs) of FIG. 3. When the scattering coefficientα is infinite, the weighting function approaches 0, and no compensatingmask may be used. In other words, when the scattering coefficient αcorresponds to α=∞, the image may be distorted minimally as illustratedin FIG. 4.

The weighting function is a function of the space frequency vectorsk_(j), k_(sim, pq), and the scattering coefficient α. The spacefrequency vectors k_(j) and k_(sim, pq) may be directly obtainedcorresponding to a specific normal mask, and the scattering coefficientα may be substituted for a predetermined value and then appropriatelyadjusted. For example, if the pattern size of a wafer is 90 nm, “2” maybe substituted for the scattering coefficient α, and the weightingfunction may thus be determined. The kernel of the multi-optical mask isthen generated and compared with a pattern size of the wafer, afterwhich the scattering coefficient α may be changed. In this manner, theweighting function varies with the scattering coefficient α. The kernelof the compensating mask may be approximately mixed with the kernel ofthe normal mask corresponding to the variation of the weightingfunction, to generate a multi-optical mask kernel.

Reference is now made to FIGS. 5A and 5C, which are graphs of pupilsurface functions of the normal and compensating masks, respectively,and FIG. 5B, which is a graph of a pupil surface function appropriatelycalibrated by reflecting the pupil surface functions of the normal andcompensating masks, according to some embodiments of the presentinvention. Each of the graphs of FIGS. 5A through 5C illustrates themagnitude and phase of a pupil surface space. Additionally, reference ismade to FIG. 6, which is a composite graph showing the graphs of FIGS.5A through 5C overlaid. As illustrated in FIG. 6, A denotes the pupilsurface function of the normal mask (FIG. 5A), C denotes the pupilsurface function of the compensating mask (FIG. 5C), and B denotes thecalibrated pupil surface function (FIG. 5B).

As shown in FIGS. 5A through 5C and 6, the effect of a high orderdiffracted light beam of the normal mask is reduced by the compensatingmask. The magnitudes of 1^(st)-order diffracted light beams to bothsides centering on a 0^(th)-order peak light beam are reduced. Thecalibration of the pupil surface function is used for mixing the kernelsof the normal and compensating masks to generate the multi-optical maskkernel.

Reference is made to FIG. 7, which shows the effect of a compensatingmask for compensating for a 3-D effect in a multi-optical systemaccording to some embodiments of the present invention. A normal mask100 having a 3-D mask effect is represented as a mixture of a normalmask 200 and a compensating mask 300, wherein the mixture canapproximate a thin mask. In this manner, the 3-D mask effect may bereduced and/or excluded. For example, a magnitude of the normal mask 100having the 3-D mask effect may be expressed using Equations 2 and 3below, such that Equation 3 is a Fourier Transform Equation of Equation2. The effect of the compensating mask on a narrow feature or a narrowpattern size may be confirmed through a Fourier Transform of a periodicmask function of one dimension, in this case the x-axis.

$\begin{matrix}{{m(x)} = {{{1/p} \cdot {{{rect}\left( {x/s} \right)} \otimes {{comb}\left( {x/p} \right)}}} - {{\sqrt{T} \cdot {1/p} \cdot {{{rect}\left\lbrack {\left( {x - {p/2}} \right)s} \right\rbrack} \otimes}}\; {{comb}\left( {x/p} \right)}} - {\sqrt{T_{new}} \cdot {1/p} \cdot {{{rect}\left\lbrack {\left( {x - {p/2}} \right)s} \right\rbrack} \otimes {{comb}\left( {x/p} \right)}}}}} & (2) \\{{\left\lbrack {m(x)} \right\rbrack} = {{1/p} \cdot \left\lbrack {\left\lbrack {s \cdot {{\sin \left( {\pi \; {k_{x} \cdot s}} \right)}/\left( {\pi \; {k_{x} \cdot s}} \right)}} \right\rbrack - {\left. \quad\left\lbrack {\omega {\sqrt{T} \cdot {{\sin \left( {\pi \; k_{x}\omega} \right)}/\left( {\pi \; k_{x}\omega} \right)} \cdot {\exp\left( {{- \pi}\; j\; k_{x}p} \right)}}} \right\rbrack \right\rbrack \cdot {\sum\limits_{\infty}^{\infty}{\sigma \left( {k_{x} - {n/p}} \right)}}} - {{1/{p\left\lbrack {\omega {\sqrt{T_{new}} \cdot {{\sin \left( {\pi \; k_{x}\omega} \right)}/\left( {\pi \; k_{x}\omega} \right)} \cdot {\exp \left( {{- \pi}\; j\; k_{x}p} \right)}}} \right\rbrack}} \cdot {\sum\limits_{\infty}^{\infty}{\sigma \left( {k_{x} - {n/p}} \right)}}}} \right.}} & (3)\end{matrix}$

The term including √{square root over (T_(new))}, reflects the effect ofthe compensating mask, p denotes a pitch, s denotes a pattern space,rect denotes a rectangular function, comb denotes δ function,

denotes a convolution computation, and T denotes a transmissioncoefficient. In general, if the magnitudes of the 0^(th)-order and1^(st)-order diffracted light beams are confirmed, the effect of thecompensating mask that corresponds to a reduction in the pitch may beconfirmed. Approximation equations of the magnitudes of the 0^(th)-orderand 1^(st)-order diffracted light beams can be expressed as Equations(4) and (5).

|MAG|₀ _(th) _(-order)≈s/p−ω√{square root over (T)}/p=[p−ω(1+√{squareroot over (T)}+√{square root over (T_(new))})]/p  (4)

|MAG|₁ _(st) _(-order)≈s/p·sinc(s/p)−ω√{square root over(T)}/p·sinc(ω/p)−ω√{square root over (T_(new))}/p·sinc(ω/p)  (5)

In Equations (4) and (5), sinc denotes the function (sinx)/x. As shownin Equations (4) and (5), the effect of the compensating mask isincreased with a reduction in the pitch p. In this regard, if a term ofthe compensating mask is not compensated for when the pitch p is small,an error increasingly occurs, causing the 3-D mask effect.

As a result, a proportion of a diffraction order is calibrated using thecompensating mask. This calibration may be performed corresponding toscattering, transmission loss, etc. of the mask on a pupil surface. Inother words, the calibration may be performed through a diffractionorder, electric field (E-field) matching, and/or the like on the pupilsurface.

Reference is now made to FIG. 8, which is a graph showing the profile ofa transmission E-field on the pupil surfaces of the normal andcompensating masks and the profile of a calibrated E-field according toa specific mask pattern in the multi-optical system of FIG. 7. Referringto FIG. 8, a proportion of an E-field or a signal is calibrated bycalibrating a diffraction order on a pupil surface. In this manner, aprofile of a finally calibrated E-field may be obtained as shown in FIG.8. MOS denotes a modified (multiple) optical system and a weightproportion is equal to the above-described weighting function. In otherwords, in the MOS, an E-field of a normal mask is mixed with an E-fieldof a compensating mask according to the weight proportion to generatethe profile of the finally calibrated E-field. The calibration of theE-field on the pupil surface corresponds to a mixture of kernels ofnormal and compensating masks, i.e. the generation of a multi-opticalsystem kernel. The pupil surface function on the graph of FIG. 8 may beregarded as the specified pupil surface function shown on the graph ofFIG. 6, and the interpretation of this is similar to that of FIG. 6.

Reference is now made to FIGS. 9A and 9B, are photographs of patternimages of the multi-optical system of FIG. 7 adopting the compensatingmask and pattern images on a wafer. FIG. 9A illustrates images of themulti-optical system (MOS). Not ee that the pattern images in adefocused state between ±0.06 μm are somewhat clear. FIG. 9B illustratespattern images formed by applying an image of the multi-optical systemto a wafer. Note that the pattern images are very clear. Additionalcalibration may be required to apply the multi-optical system in adefocused state. This may be performed after the multi-optical systemkernel is generated. Thus, the method of compensating for a 3-D maskeffect using a compensating mask allows a greater process margin and apattern of a mask approximating a desired target can be generated.

Reference is now made to FIG. 10, which is a flowchart of operations forcompensating for a 3-D mask effect using a compensating mask accordingto some embodiments of the present invention. Referring to FIG. 10, amask is loaded into an optical system (block 200). The mask may be anormal mask or a compensating mask. A near field image of the normalmask is measured (block 220). In some embodiments, the transmission andphase of the near field image of the normal mask may be measured. Insome embodiments, the transmission and phase of the near field image ofthe compensating mask may be measured (block 220 a). A standard opticalparameter is calibrated (block 230). The calibrated standard opticalparameter may be, for example, an illumination condition, an NA, awavelength λ, and/or the like. In some embodiments, the degree ofpolarization (D.O.P.) may be considered.

A pupil surface function is calibrated (block 240). Mmagnitude, phaseterms, etc. may be calibrated through the calibration of the pupilsurface function. For example, the calibration of the magnitude, phaseterms, etc. may be achieved through a diffraction order adjustment,E-field matching, etc. on a pupil surface. A kernel of the normal maskis generated (block 250). In some embodiments, an optical system towhich the normal mask is applied may be referred to as a first opticalsystem.

If the compensating mask is loaded into the optical system, a kernel ofthe compensating mask may be generated through the same process as thatby which the kernel of the normal mask is generated (block 250 a). Anoptical system to which the compensating mask is applied may referred toas a second optical system.

The kernels of the normal and compensating masks are mixed according toa weighting function to generate a multi-optical system kernel (block260). An optical system to which normal and compensating masks areapplied may be referred to as a multi-optical system. A kernel of themulti-optical system may be generated by calibrating a pupil surfacefunction corresponding to a weighting function as described above.

A normal value may be substituted for a feature size of a mask, e.g.,“2” is substituted for a feature size of 90 nm, and then applied to awafer after the multi-optical system kernel is generated, to compare theresult of the wafer with a target pattern specification. In this manner,the weighting function may be calibrated. As a result, the finalmulti-optical system kernel is completed. The calibration of theweighting function may be repeated several times to determine anaccurate weighing function. If a kernel is suitable for a specification,an appropriate mask may be fabricated and then cast usingphotolithography.

In some embodiments, OPC may be performed before the multi-opticalsystem kernel is generated. For example, in some embodiments, a 3-D maskeffect of a mask may be automatically considered during performing theOPC.

In some embodiments, restrictions on a specific pattern of an appliedwafer may be reflected on the kernel after the multi-optical systemkernel is generated. Further, as described with reference to FIGS. 9Aand 9B, calibration may be performed to apply the multi-optical systemkernel in a defocused state.

In a multi-optical system using a compensating mask and a method ofcompensating for a 3-D mask effect using the compensating mask, when OPCis performed in consideration of the 3-D mask effect, the time requiredfor performing OPC may be reduced. A compensating mask reflecting aspecific pattern may be realized as a kernel. In some embodiments, avariation of a mask caused by a topography may be reflected in anoptical system through the kernel to generate the multi-optical systemkernel, which is a mixture of the kernels of the optical system and thecompensating mask, and may approximate a thin mask. In this manner, OPCmay automatically reflect a 3-D mask effect. As a result, the timerequired for performing OPC may be much less than when performing OPCthrough a conventional accurate simulation.

As described above, a compensating mask according to the presentinvention may reduce a 3-D mask effect which may occur due to areduction in a feature size of a mask. Thus, in a multi-optical systemusing the compensating mask and a method of compensating for a 3-D maskeffect using the compensating mask, the 3-D mask effect may bediminished or eliminated. Additionally, the 3-D mask effect may beautomatically considered when applying OPC. Thus, the time required forperforming OPC may be much less than when performing OPC through anexisting accurate simulation.

Additionally, conditions which may be suitable for a specific patternand applied in a defocused state, can be reflected. Thus, a processmargin may be further achieved. As a result, a high-quality mask patternmay be fabricated.

Although the present invention has been described in terms of specificembodiments, the present invention is not intended to be limited by theembodiments described herein. Thus, the scope may be determined by thefollowing claims.

1. A mask comprising: a compensating mask in a thin mask patternconfigured to compensate for a 3-D (3-dimensional) effect caused by athickness of a normal mask used for forming a minute pattern, thecompensating mask comprising: a light transmitting layer formed of alight transmitting material; and a light blocking layer patternedunderneath the light transmitting layer and formed of a light blockingmaterial.
 2. The mask of claim 1, wherein a weighting function of thecompensating mask corresponds to a scattering coefficient α of a nearfield image of the normal mask.
 3. The mask of claim 2, wherein theweighting function is used to mix a first kernel of the normal mask witha second kernel of the compensating mask so as to generate amulti-optical system kernel.
 4. The mask of claim 3, wherein themulti-optical system kernel is applied on a wafer and a result measuredon the wafer is compared with a desired pattern specification tocalibrate the scattering coefficient α and the weighting function. 5.The mask of claim 3, wherein standard optical parameters and a pupilsurface function of a projection lens positioned under a mask of anoptical system are calibrated to generate the first kernel of the normalmask and the second kernel of the compensating mask.
 6. The mask ofclaim 5, wherein the standard optical parameters comprise anillumination condition, an NA (numeral aperture), and a wavelength, andthe pupil surface function comprises a magnitude and a phase of light asfactors.